Method and apparatus for performing localized thermal analysis and sub-surface imaging by scanning thermal microscopy

ABSTRACT

A platinum/Rhodium resistance thermal probe is used as an active device which acts both as a highly localized heat source and as a detector to perform localized differential calorimetry, by thermally inducing and detecting events such as glass transitions, meltings, recystallizations and thermal decomposition within volumes of material estimated at a few μm 3 . Furthermore, the probe is used to image variations in thermal conductivity and diffusivity, to perform depth profiling and sub-surface imaging. The maximum depth of the sample that is imaged is controlled by generating and detecting evanescent temperature waves in the sample.

The present application claims the benefit of the earlier filing date ofU.S. Provisional Application, Ser. No. 60/015,894, filed on Apr. 22,1996.

BACKGROUND

1. Field of the Invention

The present invention relates to determining thermal properties ofmaterials using a miniaturized resistive thermal probe. Moreparticularly, the present invention relates to performing localizedthermal analysis experiments whereby calorimetric information isobtained from a volume of materials on the order of a few cubic microns,whereas in conventional bulk calorimetry data is obtained from volumesof material on the order of a few cubic millimeters. The presentinvention also relates to modulating the temperature of a thermal probeto generate evanescent thermal waves in a material to thereby generatesub-surface images.

2. Background of the Invention

Several methods for the non-destructive characterization of solids makeuse of thermal excitation. Any local disruption of the structure thatresults in a change in density, specific heat, or thermal conductivitymay be detected by some type of thermal probe--often with highersensitivity than by the use of optical, X-ray, or electron-microscopetechniques. Many of these techniques use an intensity-modulated energysource to excite a sample. That is, the intensity of the energy sourceused to excite the sample is made to vary with time. Induced scatteredevanescent thermal waves are then detected by, for example, monitoringthe surface temperature of the sample. When a scanning mechanism is alsoincorporated, it is then possible to achieve spatial thermal mapping.Subsurface imaging can be performed within the depth of penetration ofthe thermal wave.

Most conventional methods of thermal imaging employ an energy beam thatemerges from a small source and spreads out according to the rules ofdiffraction. The extent of this spreading is normally governed by thewavelength associated with the energy flux. However, if the sample iswithin the "near-field" region, i.e., significantly less than onewavelength away from the source, then a greatly reduced beam diametercan be achieved. In this case, the beam diameter is not much larger thanthe size of the source itself.

This principle is applied in Scanning Probe Microscopy whereby a sharpprobe is brought in close proximity to the surface of a sample. Someprobe/sample interaction takes place. This interaction is monitored asthe probe is scanned over the surface. An image contrast is thencomputer-generated. The image contrast represents variations of someproperty (e.g., physical, mechanical, chemical) of the sample across thescanned area. One such scanning probe microscope is the Atomic ForceMicroscope (AFM). In conventional AFM, the height of a probe above thesurface being scanned is controlled by a feedback system, that keeps theforce between the probe and the surface of the sample constant. Theprobe height is monitored, and provides the data that is used to createimage contrast which represents the topography of the scanned area.

Near-field thermal imaging is described by C. C. Williams and H. K.Wickramasinghe in Photoacoustic and Photothermal Phenomena, P. Hess andJ. Peltzl (eds.), p. 364 (1988). In their device, the probe is aspecially made coaxial tip that forms a fine thermocouple junction. Thisprobe provides a spatial resolution on the order of tens of millimeters.The sample is either heated using a laser or the probe, or the sample isheated electrically. The feedback system maintains the probe temperatureconstant (instead of maintaining the force constant), by varying theprobe height as necessary.

In "Thermal Imaging Using the Atomic Force Microscope," Appl. Phys.Lett., vol. 62, pp. 2501-3 (1993), Majumdar, et al. describe a techniquefor thermal imaging that uses a simpler design of thermocouple tip, thanthat disclosed by Williams and Wickramasinghe. Majundar, et al.implemented standard atomic force microscopy feedback to maintaintip/sample contact. R. B. Dinwiddie, R. J. Pylkki and P. E. West"Thermal Conductivity Contrast Imaging with a Scanning ThermalMicroscope," Thermal Conductivity 22, T. W. Tong (ed.) (1994), describethe use of a probe in the form of a tiny platinum resistancethermometer. U.S. Pat. No. 5,441,343 to Pylkki et al., which isincorporated herein by reference, discloses the thermal sensing probefor use with a scanning probe microscope, in which the contact force ofthe probe is maintained at a constant level as the probe is scannedacross the surface of the sample.

In those studies, the samples were generally probed at a constant (ac ordc) amplitude of either surface temperature or heat flow. Thus changesin the thermal properties of materials, such as heat capacity or thermalconductivity, were not investigated. This is because the temperature ofthe sample was not raised by a sufficient amount for a change in thesample's thermal properties to be detected.

Bulk thermal analysis techniques have been developed to study suchchanges in the thermal properties of materials. Modulated temperaturedifferential scanning microscopy (MDSC) and spatially-resolved modulateddifferential scanning calorimetry (SR-MDSC) are described in U.S. Pat.No. 5,224,775 ('775 patent) and U.S. Pat. No. 5,248,199, respectively,which are both incorporated herein by reference. A conventional heatflux differential scanning calorimeter (DSC) measures the heat flow intoand out of a sample with respect to a reference. Both sample andreference are usually subjected to a linear temperature/time ramp. Inone implementation of MDSC, a sinusoidal modulation is superimposed onthe underlying heating ramp to generate a corresponding sinusoidalresponse in the heat flow signal. This results in two measurements ofheat capacity, an underlying linear long-period measurement due to theunderlying heating ramp and a higher-frequency cyclic measurement due tothe superimposed sinusoidal modulation. For many systems the cyclicmeasurement "sees" only the reversible heat capacity associated withmolecular vibrations, e.g., glass transitions, whereas the underlyingmeasurement also sees endotherms and exotherms associated withkinetically-controlled events such as recrystallizations, cure reactionsor the loss of volatile materials.

SUMMARY OF THE INVENTION

The present invention is a new analytical technique which makescalorimetric measurements on a localized scale. Data obtained from themeasurement can be used to generate contrast in our image of the thermalproperties of the sample on a localized scale. In addition, bysubjecting the sample to an oscillating program, images of the sample ata depth below the surface can be made. The depth corresponds to thefrequency of the applied oscillatory temperature.

The present invention applies modulated temperature differentialscanning calorimetry, as described in U.S. Pat. No. 5,224,755 toReading, et al. ('775 patent), which has been conventionally used toperform bulk thermal analysis experiments of a sample material, tomicroscopic thermal analysis of a sample material using two highlyminiaturized resistive probes, developed by the Topometrix Corporation(U.S. Pat. No. 5,441,343 to Pylkki et al. ('343 patent)), in adifferential configuration. A sample probe attached to a Scanning ProbeMicroscope is positioned at a desired location on the surface within thefield of view. Localized calorimetry is then performed at that positionby inducing and detecting localized phase transitions. This is achievedby ramping the temperature of the probe by passing an electrical currentthrough the probe. A small temperature oscillation is superimposed tothat temperature ramp by adding a modulated component to current theprobe current. By scanning over the surface of the sample, contrast canbe developed corresponding to particular locations on the sample tocreate an image of the thermal properties of the sample at particularlocations.

A second embodiment of the present invention allows for sub-surfaceimaging thermal microscopy to be performed by modulating the temperatureof the probe. This is done by passing a modulated current through it,thus generating thermal waves within the sample. The depth ofpenetration of these waves is frequency dependent, such that the thermalproperties of a sample can be probed as a function of depth below thesurface.

The probe, developed by the Topometrix Corporation, is an elongated loopof Wollaston wire, shaped in the form of a canteliver whose end formsthe resistive element. The resistance of that element varies withtemperature. Conversely, its temperature can be set by passing a currentof appropriate value through it. A mirror is attached across the loopallowing for the contact force of the element on the sample to be heldconstant, as in conventional atomic force microscopy, while the probe isscanned across the surface of the sample.

In the two embodiments of the present invention, the probe is used as ahighly localized heat source by passing a current through it. Thetemperature of the probe is either constant, or is variable as afunction of time. As the probe is brought close to the surface of asample, heat will flow from the probe to the sample. The amount of heatflowing will vary according to various properties of the sample at thelocation under the probe. This varying heat flow causes the temperatureof the resistive element to change, thereby changing its resistance. Afeedback circuit is preferably used to sense the change in the proberesistance (and therefore its temperature) and increase the amount ofcurrent flowing through the probe to bring it back to its originalresistance value (and therefore to its set temperature). A differentialsignal is then monitored, either directly or through a lock-inamplifier. The differential signal is used to either (1) producelocalized analysis plots of amplitude and phase data versus temperaturethat provide calorimetric information at a specific position on thesample, or (2) construct an image whose contrasts represent variationsin thermal conductivity and/or diffusivity across a scanned area. In thesecond case, the time-varying current through the resistive elementsgenerates thermal waves in the sample. The modulation frequency of thetime-varying current is functionally related to the depth below thesurface of the sample at which an image of the sample is desired. Asub-surface image is thus generated. The depth of material below thesample surface that is contributing to the image can be controlled bysuitably choosing the temperature modulation frequency. As described inAlmond, et al., "Photothermal Science and Techniques," page 15, Chapmanand Hall (London 1996), which is hereby incorporated by reference in itsentirety, the penetration depth is proportional to the square root ofthe thermal diffusivity of the sample divided by the frequency of theapplied temperature wave.

OBJECTS OF THE INVENTION

A first object of the present invention is to provide localizeddifferential thermal analysis at the micron level, such that events suchas glass transitions and meltings can be induced in a highly spatiallylocalized manner and identified.

A second object of the present invention is to use scanning thermalmicroscopy to obtain sub-surface images of solids, in which the imagecontrast is due to variations in thermal diffusivity within a givencontrollable depth.

Another object of the present invention is to use a highly miniaturizedresistive thermal probe as a highly localized source of heat, such thatit can be used to produce highly localized phase transitions at thesurface of samples.

Another object of the invention is to use modulated differentialscanning calorimetric signals to produce a high-frequency temperaturemodulation in a probe used for sensing the thermal properties ofmaterials at different depths below the surface.

Another object of the present inventions to provide an apparatus whichcan, for a sample containing different regions subject to eitherreversible or irreversible changes with temperature, produce thermalimages showing contrast based upon the reversing and nonreversing natureof the heat flow, respectively.

These and other objects of the present invention are described ingreater detail in the detailed description of the invention, theappended drawings and the attached claims.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of the cantilevered thermal resistanceprobe.

FIG. 2 is a circuit diagram of the probe control and sensing circuit.

FIGS. 3A-3C are schematic diagrams of alternative probe control andsensing circuits.

FIG. 4 is an illustration of the probe in relation to buried inclusions.

FIG. 5 is a detail of the semi-infinite volume element of FIG. 4.

FIG. 6 is a series of calculated plots of the heat flow (lower curve)and the temperature profiles (upper curves) within the sample for fivedifferent depths of a buried copper layer.

FIG. 7 is a simulated heat flow scan across two particles, buried atthree different depths (as labelled).

FIG. 8 shows the variation of the calculated lateral resolution as afunction of layer depth.

FIG. 9 shows the calculation of simulated heat flows as the probe isscanned across two particles for three different tip radii (asindicated).

FIG. 10 shows a feedback voltage line scan, for comparison with thecalculation shown in FIG. 9.

FIGS. 11A-11D are subsurface thermal images of buried particles.

FIG. 12 is a thermal image of a PVC/PB immiscible blend.

FIGS. 13A and 13B are thermal images of a PMMA/CPE blend.

FIGS. 14A and 14B illustrate temperature- and probe-induced effects ondomains in a PVC/PB blend.

FIG. 15 shows the phase signal recorded for three types of nylons, andfor PCL.

FIG. 16 is a plot of the derivative of the phase signal of PET. Theinset shows a typical heat flow signal obtained using conventional bulkdifferential scanning calorimeter.

FIGS. 17A-17E are plots of the first derivative of the phase, whichillustrate localized melting transitions for a number of polymers. Themelting temperature range as observed with conventional calorimetry isgiven in brackets as follows:

FIG. 17A: nylon 6 (210-220° C.) (the inset shows a probe current versustemperature characteristic used to achieve linearization for all thedata presented in this example).

FIG. 17B: nylon 6.6 (240-265° C.).

FIG. 17C nylon 6/10 (190-220° C.).

FIG. 17D polyethylene (130-140° C.).

FIG. 17E: polyvinylidene fluoride (155-185° C.).

FIGS. 18A-18B are plots of the first derivative of the phase versustemperature, illustrating a localized glass transition for two polymers:

FIG. 18A: polystyrene (90-110° C.).

FIG. 18B: poly(ethyl methacrylate) (60-90° C.).

FIG. 19 shows plots of the first derivative of the phase versustemperature obtained for the PEO-PS-PEO block copolymer system.

FIG. 20A shows plots of the first derivative of the phase signal versustemperature obtained at three different locations on a quenchedpoly(ethylene terephtalate) sample.

FIG. 20B are plots obtained in a temperature range in which the sampledegrades, showing three reproducible peaks.

DETAILED DESCRIPTION OF THE INVENTION

The scanning probe system used in the preferred embodiment of thepresent invention is the Explorer scanning probe microscope manufacturedby the Topometrix Corporation, located in Santa Clara, Calif.Preferably, the system is operated as a constant force microscope. Whenoperated as a force microscope, the Topometrix instrument uses a laser,together with a four-quadrant photodiode as the detection system forcantilever deflection. The scanner, probe and detection system form aself-contained system mounted independently of the sample. The sample istherefore free to be mounted on an appropriate stage (a heating stagefor example).

FIG. 1 illustrates schematically a resistive thermal probe 100 for usein a preferred embodiment of the present invention. The probe 100comprises a cantilever 102 mounted to a cantilever mount 104. The armsof the cantilever are made of Wollaston process wire having silver wirethat is preferably 75 μm in diameter. The silver wire preferablycontains a platinum/10% rhodium core. The platinum/rhodium core ispreferably 5 μm in diameter. The arms of the cantilever form a loop 106at one end of the probe 100. Where the loop 106 is formed, the silver isetched away, thereby exposing the platinum/10% rhodium core. The lengthof the exposed core is preferably 150-200 μm in length. The loop 106 isthe resistive heating source of the probe 100. As shown in FIG. 1, thecantilever 102 of the probe 100 is bent to a suitable shape, i.e., hasas sharp a bend as possible. The sharp bend yields a very small area ofcontact with the sample. The small area of contact with the sampleallows for precise heating of a particular location on the sample. Sucha probe 100 is commercially available from the Topometrix Corporation.

The probe 100 is attached to a scanning probe microscope via thecantilever mount 104. Part of the structure acts as a cantilever 102onto which a mirror 108 is cemented for the purpose of laser beamdeflection in order to achieve force control. The mirror 108 reflects alaser beam to a four-quadrant photodiode (not shown). Thelaser/mirror/photodiode system is used to sense the cantileverdeflection, and thus to provide feedback to enable the probe 100 to bescanned at constant force. The probe 100 is held magnetically in placeon the scanner in a well-known manner. An epoxy bead (not shown) isadded near the end of the cantilever to reduce the risk of breaking thefilament. Such a scanning probe system is the Explorer scanning probemicroscope produced by Topometrix Corporation.

The resistive thermal probe 100 can be used as a highly localized heatsource as well as a detector. When heated by the passage of an electriccurrent through the resistive portion 106 of the probe 100, its contactwith the sample acts as a point-like heat source. Therefore, no othermeans of sample heating, such as a laser, is required. The probe 100 isattached to a scanning mechanism, and is controlled to obtain thermalimage contrast that corresponds to variations in either thermalconductivity (using DC imaging) or thermal diffusivity (using ACimaging).

A conventional method for operating the probe is in the constanttemperature, self-heating mode. The thermal element is used as aresistive heater, as well as functioning as a temperature sensor. Asillustrated schematically in FIG. 2, the probe 100 forms one of the legsof a Wheatstone bridge 204. The remaining legs of the wheatstone bridge204 are formed by resistors R₁ and R₂, and a resistor R_(C). ResistorsR₁ and R₂ are chosen to have constant values. The resistor R_(C) ischosen to provide the set point for the bridge voltage as describedbelow. The resistor R_(L) accounts for the resistance of the probe's100's probe leads.

The Wheatstone bridge circuit 204 uses a feedback loop 205 to adjust abridge voltage V_(b) as necessary to keep the bridge 204 balanced. Bymaintaining the balance of the bridge 204, the temperature of the probeis kept constant. The feedback loop 205 generally comprises a differenceamplifier 206 and a feedback circuit 212. The difference amplifier 206has a first input 207, coupled to a junction 208 and a second input 209coupled to a junction 210. The second input 209 is the inverted input ofthe difference amplifier 206. The difference amplifier determines thevoltage difference across the junctions 208 and 210. The voltagedifference is input to the feedback circuit 212 to adjust the bridgevoltage V_(b) as required. The amount of adjustment required to maintainthe bridge balance is used to develop contrast in an image. For example,the spot intensity on a cathode ray tube (CRT) screen can be made to beproportional to the bridge adjustment voltage.

As the probe 100 contacts the sample surface 216, heat flows from theprobe 100 to the sample. In the absence of temperature feedback, thisflow of heat reduces the probe temperature, decreasing its resistanceand causing the bridge balance to shift. The feedback loop 205 sensesthis shift through difference amplifier 206 and increases the voltage,V_(b), applied to the bridge. This in turn increases the resistiveheating of the probe 100, returning its resistance to the set point. Theset point is the probe resistance that corresponds to a constant desiredoperating temperature, and is set by appropriately choosing the value ofR_(c). The probe 100 is scanned at constant force. Variations in theheat flow out of the probe 100 are measured by monitoring the bridgevoltage V_(b) required to maintain the probe's 100's resistance suchthat it produces the content desired operating temperature. The bridgevoltage signal V_(b) is then used to create contrast in the thermalimage.

Referring to FIG. 3A, a schematic diagram illustrating a first preferredembodiment of the present invention is presented. The first embodimentboth maintains a constant underlying temperature when scanning thesurface of a sample and generates thermal waves into the sample. It doesso by incorporating control circuitry 300 that adds an oscillatingvoltage signal 302 to an underlying DC voltage signal 303 to cause thetemperature of the region of the sample 216 in contact with the probe tooscillate about a fixed temperature. The fixed temperature is set by thevalue of resistor R_(c) and maintained by operation of the feedbackcircuit 306.

A summing circuit 305 is used to sum the oscillating voltage signal 302with the voltage set by the feedback circuit 306. The voltage set by thefeedback circuit 306 is the voltage that is required to maintain theprobe at a constant temperature (set by R_(c)). The addition of theoscillating voltage signal 302 generates a thermal wave in the sample216. The bridge 204 imbalance caused by the heat flow out of the probe100 to the sample is monitored using a lock-in amplifier (LIA) 304.LIAs, such as LIA 304, are well-known devices that are used for coherentdetection and recovery of modulated signals having a low signal-to-noise(SNR) ratio, i.e., that are buried in noise. Thus, the LIA 304determines the amplitude and phase of the heat flow due to theoscillating voltage signal 302. This information is subsequently used togenerate contrast in an image as described further below. The LIAcomputes the amplitude and phase (relative to a reference) of themodulated signal of interest. In the preferred embodiment, the referenceis the oscillating voltage 302. The LIA 304 of the present inventionpreferably has a frequency range from approximately 1 millihertz (mHz)to approximately 1 megahertz (MHz).

The feedback circuit 306 preferably includes a differential stage 307and an integration stage 308. The differential stage 307 determines theerror between the set point value and the value actually measured. Theerror is input to the integrator in stage 308. The integrator 308attempts to force the error to zero. Variations in the AC signal due tothe heat flow from the probe 100 to the sample 216 are used to generatecontrast in an image which represents variations in thermal diffusivityat a given temperature (set by R_(c)) within the thermal wave diffusionlength. This is accomplished by monitoring the amplitude and the phaseof the ac signal, using the LIA 304.

Using the values of the generated dc amplitude, ac amplitude and/or acphase signals, contrast is developed in an image of the sample. Theimage represents a depth of the sample determined by the frequency ofthe applied oscillating voltage 302. In the preferred embodiment, thevalue of the contrast is proportional to the value of the generated dcamplitude, ac amplitude, or ac phase signal used to generate thecontrast.

A second preferred embodiment of the present invention is illustratedschematically in FIGS. 3B and 3C. The second preferred embodimentincorporates two probes, a sample probe 320 and a reference probe 322,to perform localized thermal analysis at the surface of the sample 316.For example, localized MDSC experiments, such as those described in the'775 patent, can be performed at a location on the surface of thesample, rather than for the sample bulk as in conventional systems. Inthe preferred embodiment, both the sample probe 320 and the referenceprobe 322 are similar to the probe 100 described above.

According to the preferred embodiment of the present invention, aparticular location on the sample at which to perform a localizedthermal analysis is selected by first obtaining a thermal image of thesample. The thermal image can be obtained by a technique according tothe first embodiment of the present invention or any other technique.Using the thermal image, the particular location is selected. One methodfor selecting the particular location is to use a pointing and selectiondevice, such as a computer mouse, to select the particular location bypointing to and clicking on a particular location in a representation ofthe thermal image that is displayed on a computer monitor. The sampleprobe 320 is then positioned at the particular location by the scanningprobe microscope system. When the sample probe 320 is properlypositioned at the particular location, a thermal analysis is performedby supplying the sample probe 320 with a current to produce a heatingfunction according to a temperature program as described below.

Equal currents (I) 326 are passed into the reference probe 320 and intothe sample probe 320. An example of the currents 326 is illustratedgraphically in FIG. 3D as current 340. As shown, the current 340 has aconstant underlying heating rate on which a modulating signal issuperimposed. In the preferred embodiment, the modulating signal issinusoidal in character. However, as described in the '775 patent, themodulating signal can assume a variety of functions.

The current 340 can be generated in accordance with any desiredtemperature program. For example, temperature programs used inconventional bulk analyses can be used. In addition to the sinusoidalcurrent described above for example, the current 340 can be created bychoosing a repeating unit, and a number of repetitions for the repeatingunit. Various parameters relating to the temperature program can bedetermined from the temperature program subsequent to its generation.For example, the underlying heating rate can be determined by averagingover a single period of the temperature program. Moreover, the frequencycan be determined as the reciprocal of the period. In addition, thetemperature as measured by the sample probe 320 can be used to calculatethe underlying heating rate, and modulation amplitude and frequency asrequired. These parameters can be determined, or selected, by a computeror a user.

Referring back to FIG. 3B, the sample probe 320 is in mechanical contactwith the surface of the sample 316 at a fixed location. As describedbelow with reference to FIG. 3C, each current 326 is the sum of analternating current (ac) component and a direct current (dc) component.The ac component produces ac heating in the probes 320 and 322. Toincrease the sample temperature, the currents in the probes 320 and 320are increased accordingly. Using the present invention therefore, alocalized portion of the sample is exposed to a temperature that variesin an analogous manner to the current 340 illustrated in FIG. 3D.

The difference between the voltages across the probes is fed into alock-in-amplifier 328. A differential amplifier 330 can be used todetermine the difference. Three signals are derived using thisconfiguration. First, a DC signal is obtained after low pass filteringthrough a low pass filter (LPF) 329. The DC signal contains informationthat is related to the underlying temperature of the equal currents 326.The two remaining signals are output by the LIA 328. The LIA 328 outputsone signal that is representative of the AC amplitude of thedifferential AC signal. The LIA 328 also outputs a signal that isrepresentative of the phase of the AC differential signal. These signalscan be plotted, displayed in real time, e.g., on a CRT display, such asa computer monitor, or an oscilloscope, and/or stored to a computer diskor any other storage device.

Using these signals, the temperature at which the sample undergoes aphase transition at the particular location analyzed can be determined.The phase transition is indicated by a sharp transition or peak in oneor more of the three signals. The resulting amplitude and phase data canbe displayed or plotted in a variety of ways. For example, plots ofamplitude versus temperature and phase versus temperature can be made atspecific location on the surface of the sample. In an alternativeembodiment, layered plots of amplitude versus temperature and phaseversus temperature can be made for locations taken at even intervalsacross the surface of the sample as the surface is scanned by the sampleprobe 320.

FIG. 3C is a more detailed schematic diagram of the circuitry used toperform localized MDSC. FIG. 3C shows an integral control feedbackcircuit 332. It includes an integrator 333 and a differential amplifier330, which are used to maintain the DC temperature of the sample probe320 at the value set by the reference probe 320. The feedback circuit332 forces the difference between the voltage across the sample probe320 to be equal to the voltage across the reference probe 320 by forcingthe error between the voltages measured by the sample probe 320 and thereference probe 320 to zero. In addition, an alternating current (ac IN)is injected into the two probes to produce AC heating using the summingcircuits 334 and 336.

The bandwidth of the feedback circuit 332 can be set higher to or lowerthan the frequency of the ac modulation. If set higher, the feedbackcircuit 332 responds to both the ac and dc components of the errorsignal. If set lower, the feedback circuit 332 only responds to the dccomponent of the error signal. This flexibility provides and additionalparameter for conducting an experiment. Where the frequency is setdepends on the given experiment. Considerations as to whether to use ahigher or lower frequency include the sample material being studied andthe localized phase transition through which the sample material isdriven. Preferably, the bandwidth of the feedback circuit 332 is set to1 kHz. Furthermore, the current used is preferably such that atemperature oscillation of about one degree amplitude is obtained. Thedifferential voltage across the two probes is monitored by thedifferential amplifier 332. Three signals are recorded: the amplitude ofthe differential underlying signal (DC OUT), as measured by a summingcircuit 338, and the phase (PHASE) and amplitude (AMP) of the dynamicsignal, as measured by the lock-in amplifier 328. The DC OUT signal isfed back to the dc in input of the summing circuit 334. These threesignals can be digitized and read into a computer, or displayed in realtime, e.g., on a CRT display, such as a computer monitor, or anoscilloscope. As described above, various plots or displays of the datacan be made using these signals.

As in any macroscopic differential scanning calorimeter measurement, thetemperature ramp generally produces variations in the thermalconductivity and heat capacity of the sample, which may also undergolocal phase changes. The apparatus of FIG. 3C uses local heating ratherthan a heating stage, so that these temperature-induced variations willthemselves be subject to spatial variation according to where the sampleprobe is positioned. In turn, the heat flow out of the probe, andtherefore the resistance and the voltage across the probe, varyaccording to where the sample probe is positioned. This is reflected inthe variations of the amplitude and phase of the differential signals,as well as in the DC signal. The probed volume of material in thepresent invention is of the order of a few μm³. The probed volume is thesmallest volume that can be used to generate a useful localized MDSCscan. It can be used to map thermally activated near-surface processessuch as glass transitions, melting, cure reactions, recrystallizationsand degradations. Because the probes are used as heat sources as well asdetectors, using a single probe would give a baseline which would tendto hide any variations in the signal indicative of phase changes betweenthe applied temperature and the resulting measurement. An importantfeature of the differential arrangement is that by means of the feedbackmechanism, the sample probe is maintained at the same temperature as thereference probe.

Mathematical Modeling and Simulation of Thermal Imaging in the Case of aConstant Temperature Mode of Operation

A one-dimensional mathematical model which describes the probe-sampleinteraction has been developed in the simple case of constanttemperature (DC) thermal imaging. DC imaging can be considered as theextreme case of AC imaging when the frequency is zero. This mathematicalmodel illustrates the sub-surface imaging capability of the presentinvention.

Measurement of Heat Flow

The flow of heat in the sample can be related to the heat dissipated inthe resistive element and therefore to the voltage applied to the bridgeas follows:

The resistance R_(p) of the probe (and therefore its temperature T) isset by selecting R_(c) (see FIG. 2). This resistance can be expressedas: ##EQU1## where α is the temperature coefficient of the resistance ofthe probe material, T_(a) is the ambient temperature, R_(a) is theresistance of the probe at ambient temperature and R_(l) is the leadresistance.

When the probe is away from the sample, the energy dissipated in theprobe to raise its temperature to the desired value T is then given by:##EQU2## where V_(p0) is the voltage across the probe and is calculatedfrom the bridge voltage V_(b0) as: ##EQU3## (Part of the electricalenergy resistively dissipated in the probe will heat up a certain volumeof surrounding air).

As the probe is lowered towards the sample heat will flow out, loweringthe temperature and resistance of the probe, but the feedback circuitincreases the energy dissipated in the probe and readjusts itstemperature (and resistance) to the set value, as determined by R_(c).

The heat dissipated in the probe is now: ##EQU4## where the new voltageprobe V_(p) is calculated from the bridge feedback voltage V.

The electrical energy dissipated in the probe now also includes heatflowing into the sample. This outflow of heat can thus be measured andis given by:

Calculations--Basic one-dimensional heat flow model

    Q.sub.f =Q.sub.p -Q.sub.p0

The flow of heat in the sample is affected by several factors includingthe contact area of the probe, the temperature difference between probeand sample and the thermal conductivity of the sample. FIG. 4 is aschematic diagram depicting the probe 400 in contact with the surface ofa sample 406. The probe is modeled as a series of elemental contiguousheat sources 402, each having the same projected area of cross-sectionA. For example, elemental heat source 402 is over volume element 404.The elemental heat sources are all at the same temperature, but arelocated at different heights above the surface of the sample 406. Tosimplify the heat flow calculation, the heat flow is assumed to beunidirectional along the z-direction into the sample. As illustrated inFIG. 5, this assumption is a major simplication to the heat flow. Inreality, there is a lateral component 502 to the heat flow in eachvolume element. FIG. 5 depicts an expanded view of volume element 404.The effect of the lateral component of the heat flow 502 is accountedfor by considering the walls of the element as not being thermallyinsulating. The loss at the walls is then modeled by including a losscoefficient ε which represents heat loss through the walls of theelement 504.

Consider an element of cross-section A extending through a layer ofthickness (z₂ -Z₁) (material 2) embedded at depth z₁ in a semi-infinitematrix (material 1). The thermal conductivities of material 1 andmaterial 2 are k₁ and k₂, respectively. The corresponding part of thesurface of the thermal probe 408, which is the heat source, is at adistance z₀ from the surface. Heat conduction into the sample is througha layer of air whose thermal conductivity is k₀. The loss coefficient ε,which represents heat flow through the walls of the element, is assumedto be constant along the length of the element of volume.

In this one-dimensional approximation, the temperature profile in theheated element can be described by the following differential equation:##EQU5## with i=0, 1, 2, 3. The general solution is:

    T.sub.i (z)=P.sub.i e.sup.μ.sbsp.i.sup.z +Q.sub.i e.sup.-μ.sbsp.i.sup.z

where ##EQU6## and where P and Q are constants. The solution to theproblem is found using the following boundary conditions, which definetemperature and heat flow continuity (T_(i) is the temperature aboveambient): ##EQU7##

These equations enable the determination of the eight unknowns P_(i),Q_(i), and thus the determination of the temperature profiles T_(i) (z).The theoretical value of heat flow Q_(felm) into the heated element isgiven by: ##EQU8##

The value of the loss coefficient ε can be determined as follows. Heatflow from the probe into the sample was determined experimentally forvarious probe positions over homogeneous regions and over regions withburied inclusions of different sizes, e.g., 0.4 μm, 1.0 μm and 4.0 μm.The values of ε for each probe position are then calculated using theseexperimentally measured values of heat flow, the theoretical expressionfor the heat flow into the sample given above in equation (1), theassumed nominal values of burial depth and thickness of the inclusions,(whose thicknesses are assumed to be about one-third of the observedlateral size of the inclusions), and the values of the thermalconductivity for copper and for polystyrene shown in Table II below. Thecalculated values of ε determined at each probe position are thenaveraged to determine ε for use in subsequent calculations.

Using the calculated value of the loss coefficient ε, the total heatflow is calculated by summing the elemental heat flows along the profileof the probe, out to where the air gap was 100 μm wide. The temperatureof the probe was assumed to be constant along its length.

FIGS. 6, 7 and 8 show the results of calculations of temperatureprofiles directly below the apex of the probe, which is assumed to bejust touching the surface. The probe is assumed to have a radius ofcurvature of 20 μm and a height of 100 μm. The temperature of the probeis assumed to be uniform and constant along the wire length. Heat flowsare total heat flows from the probe and obtained by integrating alongthe shape of the probe. The profiles were calculated for the case ofcopper inclusions (thermal conductivity=400 W/m.C) embedded inpolystyrene matrix (thermal conductivity=0.13 W/m.C). The probetemperature, or T₀, was assumed to be 20° C. above ambient.

FIG. 6 shows the temperature profiles 602, 604, 606, 608, 610 and 612inside the sample for a 10 μm thick copper layer buried at depthsvarying from 0.1 to 10 μm. Because of the much higher thermalconductivity of the copper, the temperature gradient across the copperlayer is much smaller than the temperature gradient in the polystyrene.The temperature gradient refers to the rate of variation of temperaturewith depth. The lower curve 614 (right hand vertical scale) in FIG. 6 isa plot of the heat flow from the probe for the 10 μm thick copper layerversus the depth of burial of the layer. The heat flow decreases as thelayer is buried further away from the surface, and tends to the value ofheat flow for a homogeneous polystyrene sample.

FIG. 7 is a series of plots 702, 704 and 706 simulating the heat flowfrom the probe as the probe is scanned over two copper particles 1micron in length and 1 micron thick, for increasing separation andincreasing depth of burial. As a criterion of lateral resolution, weassume that two particles are resolved if: ##EQU9## where Q_(fmin) isthe minimum heat flow at a given burial depth, Q_(fmax) is the maximumheat flow at a given burial depth, Q_(f) (a) the heat flow at thetrough, and RES is a coefficient of resolution.

If RES is set equal to 20 then the graphs 802 (probe radius ρ=20 μm) and804 (probe radius ρ=1 μm) presented in FIG. 8 illustrate lateralresolution as a function of the depth at which the copper layer isburied. The graphs 802 and 804 show that the theoretical lateralresolution is on the order of a micron at the surface, but degrades whenthe particles are buried further deeply into the bulk. The insets 806and 808 illustrate alternative curves predicted by an overlapping peakscriterion (20% minimum dip in the heat flow signal between twoinclusions, neglecting the effect of noise).

Comparison to Experimental Results:

FIGS. 9 and 10 are a comparison of the simulated heat flows 902, 1002and 1004 as the probe is scanned across two particles for tip radii of10 μm, 5 μm, and 1 μm for curves 904, 906 and 908 (FIG. 9) toexperimental data (FIG. 10). The curves 904, 906 and 908 illustrate theimprovement of lateral resolution as the radius of the curvature of theprobe's tip becomes smaller.

In the scan line 1006 shown in FIG. 10, the bridge feedback voltageexcursion ranged from an average of 886 mV to an average of 897 mV. Thisline scan was obtained for the 1 μm thick coating. The feedback voltagewhen the probe was away from the surface of the sample was 784 mV. Theminimum heat flow, into a homogeneous polystyrene region, was calculatedto be equal to 165 μW and the maximum heat flow (into a region where acopper particle is buried) was 184 μW. These figures should be comparedwith theoretical values of 138 μW and 158 μW obtained for a 10 μm thickparticle buried at 1 μm depth.

Using the two experimental values of heat flow, the calculated value ofthe thermal conductivity of polystyrene was 0.21 W m⁻¹ C⁻¹ and the depthof the copper particle was 0.23μ. These discrepancies arise fromapproximations and assumptions in the model and also because the samplegeometry is not yet fully controllable. Indeed, the surfaces are nottruly flat. Furthermore, from the strength of the thermal signal it isclear that the particles were not all buried at the same depth. It islikely that when the samples were hot pressed, the particles sank atdifferent depths below the surface. The model could be extended to threedimensions, to obtain more reliable thermal data for quantitativeinterpretation. Using well-known lithography techniques, accurate andreproducible samples could be prepared for further adjustments to themodel. Such lithography techniques can be found in M. S. Tyagi,"Introduction to Semiconductor Devices," Section 19.5.2, Wiley (N.Y.1991), which is hereby incorporated by reference. A full calibration ofthe instrument would then be possible and quantitative interpretation ofthe recorded data in terms of thermal conductivity measurements, forexample, could then be achieved.

EXAMPLES

The following examples illustrate applications of the thermal probe toobtain a map of surface calorimetric data, with a lateral resolution onthe order of a micron, and a probed volume of material on the order of afew μm³. They include:

mapping at the sample surface variations in thermal conductivity and inthermal diffusivity, when the temperature-modulation mode is used;

sub-surface imaging--in principle, by varying the frequency of theevanescent temperature wave used, the depth to which the evanescentwaves penetrate (and hence the thickness of the region being imaged) canbe controlled; and

localized calorimetric analysis of micron-sized regions, as a steptowards mapping thermally activated near-surface processes such as glasstransitions, meltings, cure reactions, recrystallizations, anddegradations.

The use of the present invention in the characterization of polymerblends is of particular importance, because of the wide use of polymerblends in adhesives and coatings. In such applications, changes insurface properties as a function of temperature are of criticalimportance.

For example, film-forming emulsion polymers are widely used in papercoatings, latex paints, water-based adhesives and other applications.The properties of the film itself depend upon the way in which theindividual latex particles are able to integrate. The mechanisms of filmintegration from latex particles, and the interface development betweentwo compatible polymers are currently the subjects of significant studyby polymer scientists. For example, the properties of theparticle-particle interface affect the performance of the resultingcoating, in the case of films used in paper coating, latex paints andwater-based adhesives.

The following examples are provided to illustrate certain embodiments ofthe present invention. They are not to be construed as limiting theinvention in any way.

Example 1

This example illustrates the use of the present invention to mapheterogeneous samples with heterogeneous thermal properties. FIGS. 11A,11B, 11C and 11D were obtained with the scanning probe microscope of thepresent invention operated in the constant temperature mode and theprobe temperature set at 40° C. The thermal contrast images werecomputer-generated, using the feedback voltage V_(b) applied to theWheatstone bridge circuit 204. Because of the relatively slow scan speed(usually 100 μm s⁻¹), steady state (thermal equilibrium between theprobe and the region of the sample whose temperature is affected by theprobe) was reached at each point, so that the image contrast was indeeddetermined by the value of the heat flow. At each sampling point acertain volume of material was heated. This volume can be approximatelydelimited by the thermal contact area and an effective depth determinedby the temperature gradient below that contact area. The heat flow fromthe probe into the sample characterizes the thermal conductivities ofthe material within the heated volume.

Samples were prepared in a well-known manner by spraying polystyrenesubstrates with fine copper particles (nominally less than 1 microndiameter). The samples were then hot-pressed between glass slides at atemperature just below the melting temperature of polystyrene, and thenwere coated with a former film cast from a solution in chloroform, orcoated with layers of polystyrene films 17 μm thick (in this case theyare hot-pressed a second time).

FIGS. 11A, 11B and 11C are thermal images obtained when scanning sampleswith a coating thicknesses of 400 nm, 1 μm and 4 μm, respectively. Theyshow areas of high thermal conductivity 1102, randomly shaped, and ofsizes varying from a few microns to a few tens of microns. These figuresshow aggregates of the original particles, which have not fullydispersed below the surface of the samples. The particles are clearlydetected and show brighter in the thermal contrast, indicating that theydraw more heat from the probe, through the former film, because of thehigher thermal conductivity of the copper. Thus at each point, thesampling volume heated extends deeply enough to include the buriedparticles. However, as indicated by the scale bars, the feedback voltageexcursion diminishes as the thickness increases. Deeply buried particlesdo not show, as illustrated by FIG. 11D. Here a particle can be faintlyseen within this sample, which had been covered with a 17 μm polyesterfilm. We conclude that with a probe temperature of 40° C., the "depth ofvision" of the probe is a few microns. This is limited by the backgroundnoise (including ambient thermal fluctuations and electronic noise).

Example 2

This example demonstrates the use of the present invention for the studyof immiscible polymer blends systems, including a Poly(vinyl chloride)[PVC]/Polybutadiene [PB] blend, a Poly(ethylene oxide)[PEO]/Polybutadiene [PB] blend and a Poly(methyl methacrylate)[PMMA]/Chlorinated polyethylene [CPE] blend. The blends were cast fromsolutions on microscope glass covers with a 50/50 percentage weight.After drying, films about a hundred microns

In each system, the two polymers involved segregate, being immiscible:one forms a matrix and the other forms island-like domains. The twophases in each sample were identified by mapping the thermalconductivity variation across the surface of the sample using thethermal probe, operated in the closed loop mode at a constanttemperature of 40° C. The signal was obtained from the feedback voltageapplied to the controlling bridge in order to maintain a constant probetemperature. The contrast at each point represents variations in thermalconductivity across the sample.

FIG. 12 is a thermal image 1202 of a PVC/PB blend obtained at a zeromodulation frequency. The image contrast is a representation of thermalconductivity variations across the sample within a depth of a fewmicrons. The two phases are clearly apparent: PB, having a higherthermal conductivity (0.24 J/sec.m.K) than PVC (0.14 J/sec.m.K), isidentified by the brighter island-like domains 1204 of higher thermalconductivity. Thus in this particular system, PB segregates into domains1204 of diameter up to about 50 μm in a matrix of PVC.

FIGS. 13A and 13B are images 1302 and 1304 of a PMMA/CPE system usingunmodulated and modulated thermal probes. For FIG. 13A, the probe wasoperated in the closed loop mode at a temperature of 40° C. For FIG.13B, a 10 kHz fluctuating temperature of about 5° C. was superimposed onthe 40° C. operating temperature. In FIG. 13A, the contrast correspondsto variations in thermal conductivity. Thus in this particular system,PMMA, which has a higher thermal conductivity than CPE (0.193 against0.144 J.S⁻¹.m⁻¹.K⁻¹), segregates again into island-like domains 1306 ina matrix of CPE. In FIG. 13B, contrast arises from the phase shift ofthe AC voltage across the probe as it is scanned. The contrast in FIG.13B represents variations of thermal properties, which includediffusivity, across the sample within a depth (below the surface) equalto the thermal diffusion length, as determined by the modulationfrequency.

Example 3

This example illustrates the use of the present invention to follow, inreal time, the behavior of PVC domains in a PVC/PB blend as a functionof temperature, using a temperature-controlled hot stage. The same areawas scanned at increasing hot stage temperature, which was maintainedconstant during each scan. The probe was operated in the closed loopmode, at a constant temperature of 85° C. Images obtained at roomtemperature (RT), 50° C., 70° C., 78° C., 95° C. and 100° C. are shownin FIG. 14A (1402, 1404, 1406, 1408, 1410 and 1412). (The temperature ofthe sample is shown in the upper right-hand corner of each image.) FIG.14B also shows images (1420, 1422, 1424, 1426, 1428 and 1430) of PVCdomains in a PB matrix as a function of increasing temperature. As thetemperature is increased up to about 70° C., little change to the systemis observed, although small domains can be seen to move and are absorbedinto neighboring larger ones. As the temperature is further increasedlarge domains move outside the field of view. One domain 1414 is clearlyseen to have broken up into smaller ones 1416. The reversal in contrastin the higher temperature images is due to the temperature of the stagebecoming higher than the average temperature along the probe (set at 85°C.).

At and above room temperature, PB is in the rubbery state (the glasstransition takes place at -65° C.) and up to about 70° C., PVC is in theglassy state. The probe induces a local phase transition in PVC andfurther softening of the PB This facilitates the movement of smalldomains. Above 70° C., PVC becomes rubbery and this probably accountsfor the break up of the PVC domain under the action of the scanningprobe. Above about 90° C., PB is in an almost liquid state, and thelarge PVC domains are easily moved by the probe to beyond the field ofview.

Example 4

This example shows the use of the present invention to perform spatiallylocalized thermal analysis. Instead of using a heating stage,precisely-defined regions of the sample were heated through thetemperature range of interest. The heat was supplied by the probeitself. The probe was placed at a fixed location on the surface of asample (see FIG. 3B), and both sample probe and reference probe weresubjected to a temperature ramp with an added temperature oscillation ofabout one degree amplitude.

A number of polymeric materials were investigated. The transitions weremore sharply identified in the phase signal. Some of the resultsobtained are presented here in the form of plots of the phase, or thefirst derivative of the phase, against temperature. Data from bulksamples of the same materials obtained from conventional bulk thermalanalysis techniques (DSC and thermogravimetric analysis (TGA)) are alsopresented for comparison in Table I.

FIG. 15 shows the phase signal recorded for three types of nylons (11(1502), 6 (1504), and 6/6 (1506)) and polycaprolactone (PCL) 1508.Melting points are identified by a sharp change in the slope of thesignal. Table I shows that these changes correspond to the meltingpoints of the materials. The changes observed at higher temperatures areprobably associated with decomposition. For example, there is a goodcorrelation with the thermal decomposition temperature range obtainedfrom TGA shown in Table I.

FIG. 16 is a plot of the first derivative of the phase versustemperature obtained for quenched poly(ethylene terephtalate) (PET),covering a region of temperature over which degradation occurs. Below300° C. three events are clearly identified. These are interpreted asthe glass transition 1602, recrystallization 1604 and melting 1606,respectively, as shown in FIG. 16. The inset 1608 in FIG. 16 is atypical plot of bulk heat flow versus temperature obtained usingconventional DSC on a bulk sample from the same material. The insetshows that the same events are recorded using conventional bulk DSC.However, while the melting transition occurs over the same range oftemperature in the signal obtained using a localized scan as in the bulksignal, the glass transition and recrystallization occur at highertemperatures in the localized signal as compared with the bulk signal.This may be due to surface effects, and to the small volume of materialinvolved.

At temperatures above about 350° C., decomposition 1610 is observed.This is consistent with bulk TGA data. Furthermore the three peaks 1612observed during degradation may be associated with the breaking ofspecific chemical bonds. Degradation refers to a stage along the processwhich can eventually result in complete decomposition.

Example 5

This example shows that the phase signal can be used to identifytransitions. Some of the results obtained are presented here in the formof plots of the first derivative of the phase vs. temperature. For eachmaterial, two or three plots were obtained at two different locations onthe sample (the locations were separated by about 1 mm). These plotsdemonstrate the reproduceability of the results. These data can becompared to the data from bulk samples of the same materials obtainedfrom conventional bulk thermal analysis techniques, shown in Table I.The data described in this example were obtained by linearly increasingthe current in the probe, not the temperature of the sample. Thus therate of change of temperature is not constant. An example of atemperature/current characteristic used for data linearization is shownin the inset 1702 to FIG. 17A. Although a much faster temperature rampcan be used for the localized analysis of this example than is possiblefor bulk modulated differential scanning calorimetry, the average valueof the heating ramp was only 15° C./min, for a more reliable comparisonwith bulk data.

FIGS. 17A-17E are a series of plots illustrating localized meltingtransitions for nylon 6 (FIG. 17A, feature 1704), for nylon 6/6 (FIG.17B, feature 1706), for nylon 6/10 (FIG. 17C, feature 1708), for highdensity polyethylene (FIG. 17D, feature 1710) and for polyvinylidenefluoride (FIG. 17E, feature 1712). Melting transitions 1704, 1706, 1708,1710 and 1712 are identified by a sharp change in the slope of the phasesignal as the temperature is increased. These changes correspond to theknown melting points of the materials as obtained by conventionalcalorimetry and shown in Table 1.

FIGS. 18A-18B are plots illustrating localized glass transitions forpolystyrene (FIG. 18A, feature 1802) and for poly(ethyl methacrylate)(FIG. 18B, feature 1804). The transition to a rubbery state isidentified by a change in the slope of the phase signal, which occurs ata temperature similar to the transition temperature obtained byconventional calorimetry. However, this transition is not as pronouncedas melting transitions.

FIG. 19 shows plots obtained for the block copolymer system PEO-PS-PEO,where PEO is poly(ethylene oxide) (melting temperature in the range 60to 70° C.) and PS is polystyrene (glass transition in the range of 90 to110° C.). One transition 1902 only is observed, with no detectableseparation of transition events associated with individual polymers.This indicates that individual polymers are not present as separatechemical identities.

FIG. 20A shows plots recorded at three different locations on the samesample of quenched poly(ethylene terephtalate) (PET). Three reproducibleevents are clearly identified in each plot. These are interpreted as theglass transition 2004, recrystallization 2006 and the melting transition2008. The inset 2002 in FIG. 20A is a typical plot of bulk heat flowversus temperature obtained using conventional DSC on a sample from thesame material, illustrating the same transitions in bulk PET. FIG. 20Bshows plots 2020, 2022 and 2024 obtained for temperatures above 300° C.,covering the temperature range at which degradation occurs. Three peaks(features 2010, 2012 and 2014) are observed which are tentativelyassigned to the breaking of particular bonds.

The event associated with the glass transition in PET does not show inthe phase signal the same way as it does in the case of polystyrene(FIG. 18A, feature 1802) or in the case of poly(ethyl methacrylate)(FIG. 18B, feature 1804). This may be because when a polymer goesthrough a local phase change, such as a glass transition or a meltingtransition, it softens. This leads to changes in the area of mechanicalcontact of the probe, so that the recorded signal is affected. Thiseffect must be assessed and deconvoluted from effects due to changes inthermal conductivity/diffusivity, or to heat exchanges (endotherms andexotherms) associated with phase changes.

Further Comments

Although all the characteristic plots presented in these examples wereobtained by ramping the temperature upwards, it is also possible to usethe present invention with downward temperature ramps. For example, whenthe temperature is lowered rapidly (e.g., at 100° C./min) down throughthe melting point of a crystalline polymer (e.g., nylon 11), the changein the phase signal observed at solidification is much less pronouncedthan the change in the phase signal obtained when temperature is loweredat, e.g., 15° C./min. This may indicate that locally the material hasnot fully returned to its crystalline from, but is still (possiblypartially) amorphous. However, it is also possible that any mechanicalloading effect depends upon the direction of the temperature increase ordecrease.

Also, for a polymer at 20° C., on the basis of the rate at which thespatial resolution degrades with depth under the conditions required forthermal imaging, it can be estimated that a volume of a few cubicmicrons is probed. Of course, this volume will vary with both thetemperature and the temperature ramping rate.

The foregoing disclosure of examples and embodiments of the presentinvention has been presented for purposes of illustration anddescription. It is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Many variations andmodifications of the embodiments described herein will be obvious to oneof ordinary skill in the art in light of the above disclosure. The scopeof the invention is to be defined only by the claims appended hereto,and by their equivalents.

                  TABLE I                                                         ______________________________________                                        Characteristic Temperatures (degrees Centigrade):                                         glass             thermal                                                     transition                                                                            melting   decomposition                                   ______________________________________                                        polyethylene (high density)                                                                 -125      130 to 140                                                                              300 to 400                                  poly(ethylene terephtalate)                                                                 70 to 80  245 to 265                                                                              350 to 450                                  polystyrene    90 to 110          325-400                                     poly(methyl methacrylate)                                                                    85 to 105          240 to 350                                  poly(vinyl chloride)                                                                        65 to 85                                                        poly(phenylene oxide)                                                                       210 to 225                                                      polycaprolactone        60 to 80  275 to 400                                  poly(vinylidene floride)                                                                    -30 to -20                                                                              155 to 185                                                                              420 to 600                                  poly(ethylene oxide)                                                                         -65      60 to 70                                              polybutadiene  -95                                                            poly(ethyl methacrylate)                                                                    60 to 90                                                        nylon 11                190 to 200                                            nylon 6       40 to 60  210 to 220                                                                              350 to 450                                  nylon 6/10    44 to 55  215 to 220                                                                              400 to 450                                  nylon 6/6     50 to 60  240 to 265                                            ______________________________________                                    

                  TABLE II                                                        ______________________________________                                        Some thermophysical properties of the materials involved.                     Specific heat    Thermal    Thermal                                           capacity         conductivity                                                                             diffusivity                                       (J kg.sup.-1 K.sup.-1)                                                                         (J s.sup.-1 m.sup.-1 K.sup.-1)                                                           (× 10.sup.-6 m.sup.2 s.sup.-1)              ______________________________________                                        Polystyrene                                                                           1210         0.142      0.11                                          Copper   385         401        116                                           Air     1003         2.38 × 10                                                                          18.4                                          ______________________________________                                    

What is claimed is:
 1. A method for performing localized thermalanalysis experiments, comprising the steps of:(a) placing a sample on astage; (b) exposing a particular location on the surface of said sampleto a temperature generated in accordance with a temperature programhaving a constant component and an oscillatory component appliedsimultaneously; and (c) measuring a physical parameter indicative of athermal property of said sample at said particular location; and (d)recording a result of said measuring step (c).
 2. The method as recitedin claim 1, wherein step (b) comprises the steps of:(1) selecting anunderlying heating rate to generate a temperature ramp as said constantcomponent; (2) selecting a modulating function as said oscillatorycomponent; (3) modulating said temperature ramp with said modulatingfunction to generate said temperature program; and (4) generating saidtemperature in accordance with said temperature program.
 3. The methodas recited in claim 2, wherein step (2) comprises the step of selectinga modulation amplitude and a modulation frequency.
 4. The method asrecited in claim 3, wherein step (2) comprises the step of determiningsaid modulation frequency and said modulation amplitude from saidtemperature program.
 5. The method as recited in claim 3, wherein step(2) comprises the steps of:(i) digitizing said temperature to generate adigitized temperature; and (ii) determining said modulation frequencyand said modulation amplitude from said digitized temperature.
 6. Themethod as recited in claim 2, wherein step (2) comprises the step ofselecting a repeating unit and a number of repetitions for saidrepeating unit.
 7. The method as recited in claim 2, wherein step (1)comprises the step of selecting said underlying heating rate byaveraging over a period of said modulating function.
 8. The method asrecited in claim 2, wherein said function is a sinusoid.
 9. The methodas recited in claim 1, further comprising the step of repeating steps(a)-(d) at a plurality of particular locations on the surface of saidsample to thereby determine thermal properties of the sample at each ofsaid plurality of particular locations.
 10. The method as recited inclaim 1, further comprising the step of separating the dependence ofsaid physical parameter on said temperature into component parts. 11.The method as recited in claim 10, wherein said component partscomprises a reversing component and a non-reversing component.
 12. Themethod as recited in claim 1, wherein said physical parameter is heatflow.
 13. The method as recited in claim 1, wherein said stage is an X-Ystage.
 14. An apparatus for performing localized thermal analysisexperiments comprising:(a) a sample holder in which to place a sample;(b) means for generating a temperature by using a temperature programhaving a constant component and an oscillatory component appliedsimultaneously; (c) a sample probe to apply a temperature in accordancewith said temperature program to the sample at a particular location onthe surface of the sample; (d) monitoring means for monitoring aphysical parameter indicative of a thermal property of the sample at theparticular location; and (e) recording means for recording a signalrepresentative of said physical parameter at the particular location.15. The apparatus as recited in claim 14, wherein said monitoring meanscomprising a lock-in-amplifier to monitor a phase and amplitude of saidphysical parameter corresponding to said time-varying component.
 16. Alocalized thermal analysis instrument comprising:(a) a sample stage forholding a surface of a sample in position; (b) a probe maintained incontact with the surface of the sample, said probe functioning as aresistive heater and as a temperature sensor; (c) a feedback loop forproviding the probe with an underlying voltage signal for maintainingthe temperature of the probe at an underlying constant temperature; (d)a control circuit and a summing circuit for adding an oscillatoryvoltage signal to the underlying voltage signal for generating thermalwaves in the sample; and (e) a lock-in amplifer monitoring heat flow outof the probe to the sample and determining the amplitude and phase of aheat flow signal.
 17. The thermal analysis instrument of claim 16,wherein the probe forms one leg of a Wheatstone bridge.
 18. The thermalanalysis instrument of claim 17, wherein the lock-in amplifier monitorsthe heat flow by monitoring an imbalance in the Wheatstone bridge. 19.The thermal analysis instrument of claim 16, wherein the feedback loopcomprises an integration stage and a differential stage.
 20. The thermalanalysis instrument of claim 16, wherein the probe comprises acantilever.
 21. A method for measuring thermal properties at a specificlocation on a sample comprising:(a) positioning a tip of a thermal probein contact with the surface of the sample at the specific location onthe sample; (b) generating thermal waves in the sample by controllingthe temperature of the tip of the thermal probe such that it ismaintained according to a temperature program having a constantcomponent and an oscillatory component applied simultaneously; (c)measuring heat flow into the surface of the sample by monitoring theresistance of the thermal probe; (d) determining the amplitude and phaseof the heat flow; and (e) recording the amplitude and phase of the heatflow as a function of the temperature of the tip of the thermal probe.22. The method of claim 21, wherein the thermal probe comprises one legof a Wheatstone bridge.
 23. The method of claim 21, comprising using alock-in amplifier to monitor the resistance of the thermal probe. 24.The method of claim 21, further comprising first obtaining a thermalimage of the sample, and selecting the specific location in the thermalimage of the sample, prior to step (a).